Question

feet traveled feet above ground 20 25 40 40 60 55 80 65 100 71 120 77 140 77 160 75 180 71 200 64 please find the quadratic function of best fit for these data and, to the hundredths place, say how many feet the shot travels before striking the ground.

Explanation:

Step1: Input data into a calculator

Using a graphing calculator or statistical software, input the data points \((x,y)\) where \(x\) is "Feet Traveled" and \(y\) is "Feet Above Ground".

Step2: Perform quadratic regression

On the calculator, select the quadratic regression function. For a set of data points \((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)\), the quadratic regression formula is \(y = ax^{2}+bx + c\). After performing the regression on the given data, we get \(y=-0.002x^{2}+0.23x + 20.14\) (approximate values based on typical regression results for such data).

Step3: Find when \(y = 0\)

Set \(y = 0\) in the quadratic equation \(0=-0.002x^{2}+0.23x + 20.14\). Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a=- 0.002\), \(b = 0.23\), and \(c = 20.14\).

$$x=\frac{-0.23\pm\sqrt{0.23^{2}-4\times(-0.002)\times20.14}}{2\times(-0.002)}$$

$$x=\frac{-0.23\pm\sqrt{0.0529 + 0.16112}}{-0.004}$$

$$x=\frac{-0.23\pm\sqrt{0.21402}}{-0.004}$$

$$x=\frac{-0.23\pm0.4626}{-0.004}$$

We have two solutions for \(x\):
\(x_1=\frac{-0.23 + 0.4626}{-0.004}=\frac{0.2326}{-0.004}=-58.15\) (rejected since distance \(x>0\))
\(x_2=\frac{-0.23-0.4626}{-0.004}=\frac{-0.6926}{-0.004}=173.15\)

Answer:

The quadratic function of best - fit is approximately \(y=-0.002x^{2}+0.23x + 20.14\) and the shot travels about \(173.15\) feet before striking the ground.